New insights into the strong formulation finite element method for solving elastostatic and elastodynamic problems

References

[1] Love A.E.H., A Treatise on the Mathematical Theory of Elasticity, Dover, 1944. Search in Google Scholar

[2] Timoshenko S., Goodier J.N., Theory of Elasticity, McGraw-Hill, 1951. Search in Google Scholar

[3] Sokolnikoff I.S., Mathematical Theory of Elasticity, McGraw- Hill, 1956. Search in Google Scholar

[4] Lekhnitskii S.G., Theory of Elasticity of an Anisotropic Body, Mir Publishers, 1981. Search in Google Scholar

[5] Kaw A.K., Mechanics of CompositeMaterials, CRC Press, 1997. Search in Google Scholar

[6] Mase G.T., Mase G.E., Continuum Mechanics for Engineers, CRC Press, 1999. 10.1201/9780367803230Search in Google Scholar

[7] Jones R.M., Mechanics of Composite Materials, Taylor & Francis, 1999. Search in Google Scholar

[8] Reddy J.N., Energy Principles and Variational Methods in Applied Mechanics, John Wiley & Sons, 2002. Search in Google Scholar

[9] Wempner G., Talaslidis D., Mechanics of Solids and Shells, CRC Press, 2003. 10.1201/9780367801724Search in Google Scholar

[10] Leissa A.W., Qatu M.S., Vibrations of Continuous Systems, McGraw-Hill, 2011. Search in Google Scholar

[11] Bellman R., Casti J., Differential quadrature and long-term integration, J. Math. Anal. Appl., 1971, 34, 235-238. 10.1016/0022-247X(71)90110-7Search in Google Scholar

[12] Bellman R., Kashef B.G., Casti J., Differential quadrature: A technique for the rapid solution of nonlinear partial differential equations, J. Comput. Phys., 1972, 10, 40-52. 10.1016/0021-9991(72)90089-7Search in Google Scholar

[13] Finlayson B.A., Scriven L.E., The method of weighted residual: a review, Appl. Mech. Rev., 1966, 19, 735-748. Search in Google Scholar

[14] Gottlieb D., Orszag S.A., Numerical analysis of spectral methods. Theory and applications, CBMS-NSF Regional Conf. Ser. In Appl. Math., SIAM, 1977. 10.1137/1.9781611970425Search in Google Scholar

[15] Boyd J.P., Chebyshev and Fourier spectral methods, Dover Publications, 2001. Search in Google Scholar

[16] Quan J.R., A unified approach for solving nonlinear partial differential equations in chemical engineering applications,Master thesis, University of Nebraska-Lincoln. Search in Google Scholar

[17] Quan J.R., Chang C.T., New insights in solving distributed system equations by the quadrature method – I. Analysis, Comput. Chem. Eng., 1989, 13, 779-788. 10.1016/0098-1354(89)85051-3Search in Google Scholar

[18] Quan J.R., Chang C.T., New insights in solving distributed system equations by the quadrature method – II. Numerical experiments, Comput. Chem. Eng., 1989, 13, 1017-1024. 10.1016/0098-1354(89)87043-7Search in Google Scholar

[19] Bert C.W., Malik M., Differential quadrature method in computational mechanics, Appl. Mech. Rev., 1996, 49, 1-27. 10.1115/1.3101882Search in Google Scholar

[20] Shu C., Generalized differential-integral quadrature and application to the simulation of incompressible viscous flows including parallel computation, PhD Thesis, University of Glasgow, UK, 1991. Search in Google Scholar

[21] Shu C., Richards B.E., Parallel simulation of incompressible viscous flows by generalized differential quadrature, Comput. Syst. Eng., 1992, 3, 271-281. 10.1016/0956-0521(92)90112-VSearch in Google Scholar

[22] Shu C., Richards B.E., Application of generalized differential quadrature to solve two-dimensional incompressible Navier- Stokes equations, Int. J. Numer. Meth. Fluids, 1992, 15, 791- 798. 10.1002/fld.1650150704Search in Google Scholar

[23] Bert C.W., Jang S.K., Striz A.G., Two new approximate methods for analyzing free vibration of structural components, AIAA J., 1988, 26, 612-618. 10.2514/3.9941Search in Google Scholar

[24] Bert C.W., Jang S.K., Striz, A.G., Nonlinear bending analysis of orthotropic rectangular plates by the method of differential quadrature, Comput. Mech., 1989, 5, 217-226. 10.1007/BF01046487Search in Google Scholar

[25] Bert C.W., Malik M., Free vibration analysis of thin cylindrical shells by the differential quadrature method, J. Press. Vess. Technol., 1996, 118, 1-12. 10.1115/1.2842156Search in Google Scholar

[26] Bert C.W.,Wang X., Striz A.G., Differential quadrature for static and free vibration analyses of anisotropic plates, Int. J. Solids Struct., 1993, 30, 1737- 1744. 10.1016/0020-7683(93)90230-5Search in Google Scholar

[27] Bert C.W., Wang X., Striz A.G., Static and free vibrational analysis of beams and plates by differential quadrature method, Acta Mech., 1994, 102, 11-24. 10.1007/BF01178514Search in Google Scholar

[28] Bert C.W., Wang X., Striz A.G., Convergence of DQ method in the analysis of anisotropic plates, J. Sound Vib., 1994, 170, 140-144. 10.1006/jsvi.1994.1051Search in Google Scholar

[29] Jang S.K., Application of Differential Quadrature to the Analysis of Structural Components, PhD Thesis, University of Oklahoma, 1987. Search in Google Scholar

[30] Jang S.K., Bert C.W., Striz A.G., Application of differential quadrature to static analysis of structural components, Int. J. Numer. Meth. Engng., 1989, 28, 561-577. 10.1002/nme.1620280306Search in Google Scholar

[31] Kang K., Bert C.W., Striz A.G., Vibration analysis of shear deformable circular arches by the differential quadrature method, J. Sound Vib., 1995, 181, 353-360. 10.1006/jsvi.1995.0258Search in Google Scholar

[32] Kang K.J., Bert C.W. Striz, A.G., Vibration and buckling analysis of circular arches using DQM, Comput. Struct., 1996, 60, 49-57. 10.1016/0045-7949(95)00375-4Search in Google Scholar

[33] Shu C., Free vibration analysis of composite laminated conical shells by generalized differential quadrature, J. Sound Vib., 1996, 194, 587-604. 10.1006/jsvi.1996.0379Search in Google Scholar

[34] Shu C., An eflcient approach for free vibration analysis of conical shells, Int. J. Mech. Sci., 1996, 38, 935-949. 10.1016/0020-7403(95)00096-8Search in Google Scholar

[35] Shu C., Du H., Implementation of clamped and simply supported boundary conditions in the GDQ free vibration analysis of beams and plates, Int. J. Solids Struct., 1997, 34, 819-835. 10.1016/S0020-7683(96)00057-1Search in Google Scholar

[36] Shu C.,Wang C.M., Treatment of mixed and nonuniform boundary conditions in GDQ vibration analysis of rectangular plates, Eng. Struct., 1999, 21, 125-134. 10.1016/S0141-0296(97)00155-7Search in Google Scholar

[37] Ng T.Y., Li H., Lam K.Y., Loy C.T., Parametric instability of conical shells by the generalized differential quadrature method, Int. J. Numer. Meth. Engng., 1999, 44, 819-837. 10.1002/(SICI)1097-0207(19990228)44:6<819::AID-NME528>3.0.CO;2-0Search in Google Scholar

[38] Ng T.Y., Lam K.Y., Free vibration analysis of rotating circular cylindrical shells on elastic foundation, J. Vib. Acoust., 2000, 122, 85-89. 10.1115/1.568445Search in Google Scholar

[39] Lam K.Y., Hua L., Influence of initial pressure on frequency characteristics of a rotating truncated circular conical shell, Int. J. Mech. Sci., 2000, 42, 213-236. 10.1016/S0020-7403(98)00125-8Search in Google Scholar

[40] Shu C., Differential Quadrature and Its Application in Engineering, Springer, 2000. 10.1007/978-1-4471-0407-0Search in Google Scholar

[41] Lam K.Y., Li H., Ng T.Y., Chua C.F., Generalized differential quadrature method for the free vibration of truncated conical panels, J. Sound Vib., 2002, 251, 329-348. 10.1006/jsvi.2001.3993Search in Google Scholar

[42] Ng T.Y., Li H., Lam K.Y., Generalized differential quadrature method for the free vibration of rotating composite laminated conical shell with various boundary conditions, Int. J. Mech. Sci., 2003, 45, 567-587. 10.1016/S0020-7403(03)00042-0Search in Google Scholar

[43] Ng C.H.W., Zhao Y.B., Wei G.W., Comparison of discrete convolution and generalized differential quadrature for the vibration analysis of rectangular plates, Comput. Meth. Appl. Mech. Engng., 2004, 193, 2483-2506. 10.1016/j.cma.2004.01.013Search in Google Scholar

[44] Civalek Ö., Application of differential quadrature (DQ) and harmonic differential quadrature (HDQ) for buckling analysis of thin isotropic plates and elastic columns, Eng. Struct., 2004, 26, 171-186. 10.1016/j.engstruct.2003.09.005Search in Google Scholar

[45] Civalek Ö., Geometrically nonlinear analysis of doubly curved isotropic shells resting on elastic foundation by a combination of harmonic differential quadrature-finite difference methods, Int. J. Pres. Ves. Pip., 2005, 82, 470-479. 10.1016/j.ijpvp.2004.12.003Search in Google Scholar

[46] Viola E., Tornabene F., Vibration analysis of damaged circular arches with varying cross-section, Struct. Integr. Durab. (SIDSDHM), 2005, 1, 155-169. Search in Google Scholar

[47] Viola E., Tornabene F., Vibration analysis of conical shell structures using GDQ method, Far East J. Appl. Math., 2006, 25, 23- 39. Search in Google Scholar

[48] Tornabene F., Modellazione e Soluzione di Strutture a Guscio in Materiale Anisotropo, PhD Thesis, University of Bologna, Italy, 2007. Search in Google Scholar

[49] Tornabene F., Viola E., Vibration analysis of spherical structural elements using the GDQ method, Comput. Math. Appl., 2007, 53, 1538-1560. 10.1016/j.camwa.2006.03.039Search in Google Scholar

[50] Viola E., Dilena M., Tornabene F., Analytical and numerical results for vibration analysis of multi-stepped and multidamaged circular arches, J. Sound Vib., 2007, 299, 143-163. 10.1016/j.jsv.2006.07.001Search in Google Scholar

[51] Marzani A., Tornabene F., Viola E., Nonconservative stability problems via generalized differential quadrature method, J. Sound Vib. 315, 2008, 176-196. 10.1016/j.jsv.2008.01.056Search in Google Scholar

[52] Tornabene F., Viola E., 2-D solution for free vibrations of parabolic shells using generalized differential quadrature method, Eur. J. Mech. A-Solid, 2008, 27, 1001-1025. 10.1016/j.euromechsol.2007.12.007Search in Google Scholar

[53] Tornabene F., Free vibration analysis of functionally graded conical, cylindrical and annular shell structures with a fourparameter power-law distribution, Comput. Methods Appl. Mech. Engrg., 2009, 198, 2911-2935. 10.1016/j.cma.2009.04.011Search in Google Scholar

[54] Tornabene F., Viola E., Free vibrations of four-parameter functionally graded parabolic panels and shell of revolution, Eur. J. Mech. A-Solid, 2009, 28, 991-1013. 10.1016/j.euromechsol.2009.04.005Search in Google Scholar

[55] Tornabene F., Viola E., Free vibration analysis of functionally graded panels and shells of revolution, Meccanica, 2009, 44, 255-281. 10.1007/s11012-008-9167-xSearch in Google Scholar

[56] Hong C.C., Thermal bending analysis of shear-deformable laminated anisotropic plates by the GDQ method, Mech. Res. Commun., 2009, 36, 804-810. 10.1016/j.mechrescom.2009.05.007Search in Google Scholar

[57] Tornabene F., Viola E., Inman D.J., 2-D differential quadrature solution for vibration analysis of functionally graded conical, cylindrical and annular shell structures, J. Sound Vib., 2009, 328, 259-290. 10.1016/j.jsv.2009.07.031Search in Google Scholar

[58] Viola E., Tornabene F., Free vibrations of three parameter functionally graded parabolic panels of revolution, Mech. Res. Commun., 2009, 36, 587-594. 10.1016/j.mechrescom.2009.02.001Search in Google Scholar

[59] Sadeghian H., Rezazadeh G., Comparison of generalized differential quadrature and Galerkin methods for the analysis of micro-electro-mechanical coupled systems, Communications in Nonlinear Science and Numerical Simulation, 2009, 14, 2807-2816. 10.1016/j.cnsns.2008.07.016Search in Google Scholar

[60] Tornabene F., Marzani A., Viola E., Elishakoff I., Critical flow speeds of pipes conveying fluid by the generalized differential quadrature method, Adv. Theor. Appl. Mech., 2010, 3, 121-138. Search in Google Scholar

[61] Tornabene F., 2-D GDQ solution for free vibrations of anisotropic doubly-curved shells and panels of revolution, Compos. Struct., 2011, 93, 1854-1876. 10.1016/j.compstruct.2011.02.006Search in Google Scholar

[62] Tornabene F., Free vibrations of anisotropic doubly-curved shells and panels of revolution with a free-form meridian resting on Winkler-Pasternak elastic foundations, Compos. Struct., 2011, 94, 186-206. 10.1016/j.compstruct.2011.07.002Search in Google Scholar

[63] Tornabene F., Liverani A., Caligiana G., FGM and laminated doubly-curved shells and panels of revolution with a free-form meridian: a 2-D GDQ solution for free vibrations, Int. J. Mech. Sci., 2011, 53, 446-470. 10.1016/j.ijmecsci.2011.03.007Search in Google Scholar

[64] Viola E., Rossetti L., Fantuzzi N., Numerical investigation of functionally graded cylindrical shells and panels using the generalized unconstrained third order theory coupled with the stress recovery, Compos. Struct., 2012, 94, 3736-3758. 10.1016/j.compstruct.2012.05.034Search in Google Scholar

[65] Tornabene F., Liverani A., Caligiana G., Laminated composite rectangular and annular plates: a GDQ solution for static analysis with a posteriori shear and normal stress recovery, Compos. Part B-Eng., 2012, 43, 1847-1872. 10.1016/j.compositesb.2012.01.065Search in Google Scholar

[66] Tornabene F., Liverani A., Caligiana G., Static analysis of laminated composite curved shells and panels of revolution with a posteriori shear and normal stress recovery using generalized differential quadrature method, Int. J. Mech. Sci., 2012, 61, 71- 87. 10.1016/j.ijmecsci.2012.05.007Search in Google Scholar

[67] Tornabene F., Liverani A., Caligiana G., General anisotropic doubly-curved shell theory: a differential quadrature solution for free vibrations of shells and panels of revolutionwith a freeform meridian, J. Sound Vib., 2012, 331, 4848-4869. 10.1016/j.jsv.2012.05.036Search in Google Scholar

[68] Ferreira A.J.M., Viola E., Tornabene F., Fantuzzi N., Zenkour A.M., Analysis of sandwich plates by generalized differential quadrature method, Math. Probl. Eng., 2013, 2013, 1-22, Article ID 964367. 10.1155/2013/964367Search in Google Scholar

[69] Tornabene F., Ceruti A., Free-form laminated doubly-curved shells and panels of revolution resting on Winkler-Pasternak elastic foundations: a 2-D GDQ solution for static and free vibration analysis, World J. Mech., 2013, 3, 1-25. 10.4236/wjm.2013.31001Search in Google Scholar

[70] Tornabene F., Reddy J.N., FGM and laminated doubly-curved and degenerate shells resting on nonlinear elastic foundations: a GDQ solution for static analysiswith a posteriori stress and strain recovery, J. Indian Inst. Sci., 2013, 93, 635-688. Search in Google Scholar

[71] Tornabene F., Viola E., Static analysis of functionally graded doubly-curved shells and panels of revolution, Meccanica, 2013, 48, 901-930. 10.1007/s11012-012-9643-1Search in Google Scholar

[72] Tornabene F., Viola E., Fantuzzi N., General higher-order equivalent single layer theory for free vibrations of doubly-curved laminated composite shells and panels, Compos. Struct., 2013, 104, 94-117. 10.1016/j.compstruct.2013.04.009Search in Google Scholar

[73] Viola E., Tornabene F., Fantuzzi N., General higher-order shear deformation theories for the free vibration analysis of completely doubly-curved laminated shells and panels, Compos. Struct., 2013, 95, 639-666. 10.1016/j.compstruct.2012.08.005Search in Google Scholar

[74] Viola E., Tornabene F., Fantuzzi N., Static analysis of completely doubly-curved laminated shells and panels using general higher-order shear deformation theories, Compos. Struct., 2013, 101, 59-93. 10.1016/j.compstruct.2013.01.002Search in Google Scholar

[75] Tornabene F., Fantuzzi N., Mechanics of Laminated Composite Doubly-Curved Shell Structures. The Generalized Differential Quadrature Method and the Strong Formulation Finite Element Method, Esculapio, 2014. Search in Google Scholar

[76] Ferreira A.J.M., Carrera E., Cinefra M., Viola E., Tornabene F., Fantuzzi N., Zenkour A.M., Analysis of thick isotropic and cross-ply laminated plates by generalized differential quadrature method and a unified formulation, Compos. Part B-Eng., 2014, 58, 544-552. 10.1016/j.compositesb.2013.10.088Search in Google Scholar

[77] Tornabene F., Fantuzzi N., Viola E., Carrera E., Static analysis of doubly-curved anisotropic shells and panels using CUF approach, differential geometry and differential quadrature method, Compos. Struct., 2014, 107, 675-697. 10.1016/j.compstruct.2013.08.038Search in Google Scholar

[78] Tornabene F., Fantuzzi N., Viola E., Reddy J.N., Winkler- Pasternak foundation effect on the static and dynamic analyses of laminated doubly-curved and degenerate shells and panels, Compos. Part B-Eng., 2014, 57, 269-296. 10.1016/j.compositesb.2013.06.020Search in Google Scholar

[79] Tornabene F., Ceruti, A., Mixed static and dynamic optimization of four-parameter functionally graded completely doublycurved and degenerate shells and panels using GDQ method, Math. Probl. Eng., 2013, 2013, 1-33, Article ID 867079. 10.1155/2013/867079Search in Google Scholar

[80] Viola E., Rossetti L., Fantuzzi N., Tornabene F., Static analysis of functionally graded conical shells and panels using the generalized unconstrained third order theory coupled with the stress recovery, Compos. Struct., 2014, 112, 44-65. 10.1016/j.compstruct.2014.01.039Search in Google Scholar

[81] Tornabene F., Fantuzzi N., Bacciocchi M., The local GDQ method applied to general higher-order theories of doublycurved laminated composite shells and panels: the free vibration analysis, Compos. Struct., 2014, 116, 637-660. 10.1016/j.compstruct.2014.05.008Search in Google Scholar

[82] Tornabene F., Fantuzzi N., Bacciocchi M., Free vibrations of free-form doubly-curved shells made of functionally graded materials using higher-order equivalent single layer theories, Compos. Part B Eng., 2014, 67, 490-509. 10.1016/j.compositesb.2014.08.012Search in Google Scholar

[83] Viola E., Tornabene F., Fantuzzi N., Stress and strain recovery of laminated composite doubly-curved shells and panels using higher-order formulations, Key Eng. Mat., 2015, 624, 205-213. 10.4028/www.scientific.net/KEM.624.205Search in Google Scholar

[84] Tornabene F., Fantuzzi N., Viola E., Batra R.C., Stress and strain recovery for functionally graded free-form and doubly-curved sandwich shells using higher-order equivalent single layer theory, Compos. Struct., 2015, 119, 67-89. 10.1016/j.compstruct.2014.08.005Search in Google Scholar

[85] Liu G.R., Mesh Free Methods. Moving Beyond the Finite Element Method, CRC Press, 2003. 10.1201/9781420040586Search in Google Scholar

[86] Li H.,Mulay S.S., Meshless Methods and their Numerical Properties, CRC Press, 2013. 10.1201/b14492Search in Google Scholar

[87] Liew K.M., Huang Y.Q., Reddy J.N., Analysis of general shaped thin plates by the moving least-squares differential quadrature method, Finite Elem. Anal. Design, 2004, 40, 1453-1474. 10.1016/j.finel.2003.10.002Search in Google Scholar

[88] Li Q.S., Huang Y.Q., Moving least-squares differential quadrature method for free vibration of antisymmetric laminates, J. Eng. Mech., 2004, 130, 1447-1457. 10.1061/(ASCE)0733-9399(2004)130:12(1447)Search in Google Scholar

[89] Huang Y.Q., Li Q.S., Bending and buckling analysis of antisymmetric laminates using the moving least square differential quadrature method, Comput. Meth. Appl. Mech. Engng., 2004, 193, 3471-3492. 10.1016/j.cma.2003.12.039Search in Google Scholar

[90] Lanhe W., Hua L., Daobin W., Vibration analysis of generally laminated composite plates by the moving least squares differential quadrature method, Compos. Struct., 2005, 68, 319- 330. 10.1016/j.compstruct.2004.03.025Search in Google Scholar

[91] Sator L., Sladek V., Sladek J., Coupling effects in elastic analysis of FGM composite plates by mesh-free methods", Compos. Struct., 2014, 115, 100-110. 10.1016/j.compstruct.2014.04.016Search in Google Scholar

[92] Sator L., Sladek V., Sladek J., Analysis of Beams with Transversal Gradations of the Young’s Modulus and Variable Depths by the Meshless Method, Slovak J. Civil Eng., 2014, 22, 23-36. 10.2478/sjce-2014-0004Search in Google Scholar

[93] Belytschko T., Lu Y.Y., Gu L., Element-free Galerkin methods, Int. J. Numer. Meth. Engng., 1994, 37, 229-256. 10.1002/nme.1620370205Search in Google Scholar

[94] Belytschko T., Lu Y.Y., Gu L., Tabbara M., Element-free Galerkin methods for static and dynamic fracture, Int. J. Solids Struct., 1995, 32, 2547- 2570. 10.1016/0020-7683(94)00282-2Search in Google Scholar

[95] Belytschko T., Krongauz Organ D., Fleming M., Krysl P., Meshless methods: an overview and recent developments, Comput. Meth. Appl. Engng., 1996, 139, 3-47. 10.1016/S0045-7825(96)01078-XSearch in Google Scholar

[96] Krysl P., Belytschko T., Element-free Galerkin method: convergence, Comput. Meth. Appl. Mech. Engng., 1997, 148, 257-277. 10.1016/S0045-7825(96)00007-2Search in Google Scholar

[97] Atluri S.N., Zhu T., A new Meshless Local Petrov-Galerkin (MLPG) approach to nonlinear problems in computer modeling & simulation, Comput. Model. Simul. Eng., 1998, 3, 187-196. Search in Google Scholar

[98] Atluri S.N., Zhu T., A new Meshless Local Petrov-Galerkin (MLPG) approach in computational mechanics, Comput. Mech., 1998, 22, 117-127. 10.1007/s004660050346Search in Google Scholar

[99] Atluri S.N., Zhu T., New concepts in meshless methods, Int. J. Numer. Meth. Engng., 2000, 47, 537-556. 10.1002/(SICI)1097-0207(20000110/30)47:1/3<537::AID-NME783>3.0.CO;2-ESearch in Google Scholar

[100] Atluri S.N., Zhu T.L., The meshless local Petrov-Galerkin (MLPG) approach for solving problems in elasto-statics, Comput. Mech., 2000, 25, 169-179. 10.1007/s004660050467Search in Google Scholar

[101] Kansa E.J., Multiquadrics - A scattered data approximation scheme with applications to computational fluid-dynamics - I surface approximations and partial derivative estimates, Comput. Math. Appl., 1990, 19, 127-145. 10.1016/0898-1221(90)90270-TSearch in Google Scholar

[102] Kansa E.J., Multiquadrics - A scattered data approximation scheme with applications to computational fluid-dynamics - II solutions to parabolic, hyperbolic and elliptic partial differential equations, Comput. Math. Appl., 1990, 19, 147-161. 10.1016/0898-1221(90)90271-KSearch in Google Scholar

[103] Fasshauer G.E., Solving differential equationswith radial basis functions: multilevel methods and smoothing”, Adv. Comput. Math., 1999, 11, 139-159. Search in Google Scholar

[104] Ferreira A.J.M., Fasshauer G.E., Computation of natural frequencies of shear deformable beams and plates by an RBFpseudospectral method, Comput. Meth. Appl. Mech. Engng., 2006, 196, 134-146. 10.1016/j.cma.2006.02.009Search in Google Scholar

[105] Ferreira A.J.M., Fasshauer G.E., Batra R.C., Rodrigues J.D., Static deformations and vibration analysis of composite and sandwich plates using a layerwise theory and RBF-PS discretizations with optimal shape parameter, Compos. Struct., 2008, 86, 328-343. 10.1016/j.compstruct.2008.07.025Search in Google Scholar

[106] Wendland H., Meshless Galerkin methods using radial basis functions, Math. Comput., 1999, 68, 1521- 1531. 10.1090/S0025-5718-99-01102-3Search in Google Scholar

[107] Buhmann M.D., Radial basis functions, Acta Numer., 2000, 9, 1-38. 10.1017/S0962492900000015Search in Google Scholar

[108] Shu C., Ding H., Yeo K.S., Solution of partial differential equations by a global radial basis function- based differential quadrature method, Eng. Anal. Bound. Elem., 2004, 28, 1217- 1226. 10.1016/j.enganabound.2003.02.001Search in Google Scholar

[109] Tornabene F., Fantuzzi N., Viola E., Ferreira A.J.M., Radial Basis Function method applied to doubly-curved laminated composite shells and panels with a general higher-order equivalent single Layer theory, Compos. Part B-Eng., 2013, 55, 642-659. 10.1016/j.compositesb.2013.07.026Search in Google Scholar

[110] Civan F., Sliepcevich C.M., Application of differential quadrature in solution of pool boiling in cavities, Proc. Oklahoma Acad. Sci., 1985, 65, 73-78. Search in Google Scholar

[111] Chen W.L., A New Approach for Structural Analysis. The Quadrature Element Method, PhD Thesis, University of Oklahoma, 1994. Search in Google Scholar

[112] ChenW.L., Striz A.G., Bert C.W., High accuracy plane stress and plate element in the quadrature element method, Int. J. Solids Struct., 2000, 37, 627- 647. 10.1016/S0020-7683(99)00028-1Search in Google Scholar

[113] Zhong H., He Y., Solution of Poisson and Laplace equations by quadrilateral quadrature element. Int. J. Solids Struct., 1998, 35, 2805-2819. 10.1016/S0020-7683(97)00277-1Search in Google Scholar

[114] Shu C., Chew Y.T., Khoo B.C., Yeo K.S., Application of GDQ scheme to simulate incompressible viscous flows around complex geometries, Mech. Res. Commun., 1995, 22, 319-325. 10.1016/0093-6413(95)00031-LSearch in Google Scholar

[115] Shu C., Chew Y.T., Liu Y., An eflcient approach for numerical simulation of flows in Czochralski crystal growth, J. Cryst. Growth, 1997, 181, 427- 436. 10.1016/S0022-0248(97)00296-0Search in Google Scholar

[116] Lam S.S.E., Application of the differential quadrature method to two-dimensional problems with arbitrary geometry. Comput. Struct., 1993, 47, 459-464. 10.1016/0045-7949(93)90241-5Search in Google Scholar

[117] Bert C.W., Malik M., The differential quadrature method for irregular domains and application to plate vibration, Int. J. Mech. Sci., 1996, 38, 589-606. 10.1016/S0020-7403(96)80003-8Search in Google Scholar

[118] Chen C.-N., A generalized differential quadrature element method. Comput. Meth. Appl. Mech. Engrg, 2000, 188, 553- 566. 10.1016/S0045-7825(99)00283-2Search in Google Scholar

[119] Chen C.-N., DQEM and DQFDM for the analysis of composite two-dimensional elasticity problems, Compos. Struct., 2013, 59, 3-13. 10.1016/S0263-8223(02)00231-3Search in Google Scholar

[120] Zhang Y.Y., Development of Differential Quadrature Methods and Their Applications to Plane Elasticity, PhD Thesis, National University of Singapore, 2003. Search in Google Scholar

[121] Zong Z., Lam K., Zhang Y., A multidomain differential quadrature approach to plane elastic problems with material discontinuity, Math. Comput. Model., 2005, 41, 539-553. 10.1016/j.mcm.2003.11.009Search in Google Scholar

[122] Zong Z., Zhang Y.Y., Advanced Differential Quadrature Methods, CRC Press, 2009. 10.1201/9781420082494Search in Google Scholar

[123] Xing Y., Liu B., High-accuracy differential quadrature finite element method and its application to free vibrations of thin plate with curvilinear domain, Int. J. Numer. Meth. Eng., 2009, 80, 1718-1742. 10.1002/nme.2685Search in Google Scholar

[124] Xing Y., Liu B., Liu G., A differential quadrature finite element method, Int. J. Appl. Mech., 2010, 2, 207-227. 10.1142/S1758825110000470Search in Google Scholar

[125] Zhong H., Yu T., A weak form quadrature element method for plane elasticity problems, Appl.Math. Model., 2009, 33, 3801- 3814. 10.1016/j.apm.2008.12.007Search in Google Scholar

[126] Fantuzzi N., Generalized differential quadrature finite element method applied to advanced structural mechanics, PhD Thesis, University of Bologna, 2013. Search in Google Scholar

[127] Viola E., Tornabene F., Fantuzzi N., Generalized differential quadrature finite element method for cracked composite structures of arbitrary shape, Compos. Struct., 2013, 106, 815-834. 10.1016/j.compstruct.2013.07.034Search in Google Scholar

[128] Viola E., Tornabene F., Ferretti E., Fantuzzi N., Soft core plane state structures under static loads using GDQFEM and Cell method, CMES, 2013, 94, 301-329. Search in Google Scholar

[129] Viola E., Tornabene F., Ferretti E., Fantuzzi N., GDQFEM numerical simulations of continuous media with cracks and discontinuities, CMES, 2013, 94, 331-368. Search in Google Scholar

[130] Viola E., Tornabene F., Ferretti E., Fantuzzi N., On static analysis of plane state structures via GDQFEM and Cell method, CMES, 2013, 94, 421-458. Search in Google Scholar

[131] Fantuzzi N., Tornabene F., Strong formulation finite element method for arbitrarily shaped laminated plates - I. Theoretical analysis, Adv. Aircraft Space. Sci., 2014, 1, 124-142. 10.12989/aas.2014.1.2.125Search in Google Scholar

[132] Fantuzzi N., Tornabene F., Strong formulation finite element method for arbitrarily shaped laminated plates - II. Numerical analysis, Adv. Aircraft Space. Sci., 2014, 1, 143-173. 10.12989/aas.2014.1.2.125Search in Google Scholar

[133] Fantuzzi N., Tornabene F., Viola E., Generalized differential quadrature finite element method for vibration analysis of arbitrarily shaped membranes, Int. J. Mech. Sci., 2014, 79, 216- 251. 10.1016/j.ijmecsci.2013.12.008Search in Google Scholar

[134] Fantuzzi N., Tornabene F., Viola E., Ferreira A.J.M., A Strong Formulation Finite Element Method (SFEM) based on RBF and GDQ techniques for the static and dynamic analyses of laminated plates of arbitrary shape, Meccanica, 2014, 49, 2503-2542. 10.1007/s11012-014-0014-ySearch in Google Scholar

[135] Fantuzzi N., Tornabene F., Viola E., Four-parameter functionally graded cracked plates of arbitrary shape: a GDQFEM solution for free vibrations, Mech. Adv. Mat. Struct., 2014, (in press), DOI: 10.1080/15376494.2014.933992Search in Google Scholar

[136] Tornabene F., Fantuzzi N., Bacciocchi M., The strong formulation finite element method: stability and accuracy, Fract. Struct. Integr., 2014, 29, 251-265. 10.3221/IGF-ESIS.29.22Search in Google Scholar

[137] Tornabene F., Fantuzzi N., Ubertini F., Viola E., Strong formulation finite element method based on differential quadrature: a survey, Appl. Mech. Rev., 2014, (in press), DOI:10.1115/1.4028859 10.1115/1.4028859Search in Google Scholar

[138] Zhong H., He Y., A note on incorporation of domain decomposition into the differential quadrature method, Commun. Numer. Methods Engrg., 2003, 19, 297-306. 10.1002/cnm.591Search in Google Scholar

[139] Bardell N.S., Langley R.S., Dunsdon J.M., On the free in-plane vibration of isotropic rectangular plates, J. Sound Vib., 1996, 191, 459-467. 10.1006/jsvi.1996.0134Search in Google Scholar

[140] Hyde K., Chang J.Y., Bacca C., Wickert J.A., Parameter studies for plane stress in-plane vibration of rectangular plates, J. Sound Vib., 2001, 247, 471-487. 10.1006/jsvi.2001.3767Search in Google Scholar

[141] Singh A.V., Muhammad T., Free in-plane vibration of isotropic non-rectangular plates, J. Sound Vib., 2004, 273, 219-231. 10.1016/S0022-460X(03)00496-6Search in Google Scholar

[142] Park C.I., Frequency equation for the in-plane vibration of a clamped circular plate, J. Sound Vib., 2008, 313, 325-333. 10.1016/j.jsv.2007.11.034Search in Google Scholar

[143] Bashmal S., Bhat R., Rakheja S., In-plane free vibration of circular annular disks, J. Sound Vib., 2009, 322, 216-226. 10.1016/j.jsv.2008.11.024Search in Google Scholar

[144] R.H. Macneal, R.L. Harder, A proposed standard set of problems to test finite element accuracy. Finite Elem. Anal Design, 1985, 1, 3-20. 10.1016/0168-874X(85)90003-4Search in Google Scholar

[145] Rezaiee-Pajand, M. Karkon, An effective membrane element based on analytical solution, Eur. J. Mech. A/Solids, 2013, 39, 268-279. 10.1016/j.euromechsol.2012.12.004Search in Google Scholar

[146] M.C. Amirani, S. Khalili, N. Nemati, Free vibration analysis of sandwich beam with FG core using the element free Galerkin method, Compos. Struct.,2009, 90, 373-379. 10.1016/j.compstruct.2009.03.023Search in Google Scholar

[147] R. Cook, Concepts and applications of finite element analysis. Wiley, 2001. Search in Google Scholar

[148] R. Cook, J. Avrashi, Error estimation and adaptive meshing for vibration problems, Comput. Struct., 1992, 44, 619-626. 10.1016/0045-7949(92)90394-FSearch in Google Scholar

[149] K.K. Gupta, Development of a finite dynamic element for free vibration analysis of two-dimensional structures. Int. J. Numer. Methods Engrg., 1978, 12, 1311-1327. 10.1002/nme.1620120808Search in Google Scholar

[150] Y. Li, N. Fantuzzi, F. Tornabene, On mixed mode crack initiation and direction in shafts: strain energy density factor and maximum tangential stress criteria, Eng. Fract. Mech., 2013, 109, 273-289. 10.1016/j.engfracmech.2013.07.008Search in Google Scholar

[151] Y. Li, E. Viola, Size effect investigation of a central interface crack between two bonded dissimilar materials. Compos. Struct., 2013, 105, 90-107. 10.1016/j.compstruct.2013.05.003Search in Google Scholar

[152] E. Viola, N. Fantuzzi, A. Marzani, A. (2012): Cracks interaction effect on the dynamic stability of beams under conservative and nonconservative forces. Key Eng. Mat., 2012, 488-489, 383-386. Search in Google Scholar

[153] E. Viola, Y. Li, N. Fantuzzi, N. (2012): On the stress intensity factors of cracked beams for structural analysis. Key Eng. Mat., 2012, 488-489, 379-382. Search in Google Scholar